Mad scientist Muscle Length and Tension Experiment

https://web.archive.org/web/20220710184230/https://en.wikipedia.org/wiki/Hill's_muscle_model

Similar but probably not identical studies have been performed putting electricity through frog muscles.  I do not endorse the doing the following experiment on humans without their consent nor on other living animals that can not consent nor killing animals to perform this experiment for ethical reasons.  This is a fictitious thought experiment not a experiment that should be performed in real life.

Let's say there is a mad scientist and he wants to know how the position of a nearby joint a muscle does not cross effects the length and tension of that muscle.  

In order to simplify things he decides to take muscles out of a living body and attach them to various objects with tendons while using mad science to provide a constant supply of nutrition and an electrical source.  He can adjust the current and voltage waveform to anything he wants at any time he wants.  He also let's the muscle have it's own source of various nerves that are somehow able to function and send electrical current to the muscle for various reflexes which work like muscle spindles and golgi tendon organs.  Basically these muscles work like regular muscles except they are outside the human body so he can attach them to objects in any configuration he wants and he can stimulate them with an external source of electricity.  Each and every muscle he uses is identical in resting length, resting shape, resting thickness, mass and physiology

He knows muscles do not work like springs because a spring has a specific length and only one length it has for each amount of tensile force where as a muscle has a minimum and maximum tensile force it can experience at each length.  

First Experiment muscle length measured for different amounts of weight and electricity

He attaches a single muscle to a rigid ceiling on the top and a weight on the bottom.  

He notices that when the electrical current and voltage he applies to it is the same when the weight is made more heavy the muscle is longer but when the weight is made less heavy the muscle is shorter

When the weight attached to the muscle is kept a constant amount no matter how much or how little electricity is applied the muscle tension is the same but the muscle length changes.  

Second Experiment isometric maximum isometric muscle strength and minimum muscle tension measured at different muscle lengths

He attaches a single muscle horizontally to two spring gauges that are attached to two immovable walls.  These spring gauges are extraordinary spring gauges in that even when the muscle exerts at maximum force the amount of length of the spring changes so little that it is negligible yet it is sufficient to get a accurate and precise enough reading of the muscle force.  No matter how strongly the muscle contracts pulling the spring gauges the muscle length which is the distance between the two spring gauges remains the same for all practical purposes.

He does this many different times with different lengths the walls are apart

He starts with walls at resting muscle length apart for the shortest distance between walls

And for the longest distance between walls he sets the distance where the muscle no longer produces measurable increase in force when zapped with electricity or where he feels the muscle would break if stretched any farther whichever is a shorter distance between walls

He tries many distances between walls in between those two values

He finds that the left spring gauge always measures the same force as the right spring gauge

He finds that when no external source of electricity is supplied to the muscle other than that from it's own nervous system the farther apart the walls are the greater amount of force measured on the spring gauges

He finds that when he sets the electricity to produce the highest force contraction he can for each distance between walls.  Then the farther apart he sets the walls the lower the maximum force the spring gauges read.

Third experiment muscles attached in series

He uses three string gauges, two walls, and two muscles.  He attaches one muscle to the string guage on the left wall and one muscle to the string gauge on the right wall.  The third string gauge is attached to both muscles and is between the two walls and not attached to any wall.

He finds that all three string gauges always measure the same value as each other

He finds that when the amount of electricity he sends to both muscles is identical both muscles are always at a equal length.  But, he can change the amount of force that all three string gauges identically experience by changing the amount of electricity he identically supplies to both muscles.

He finds that when he sends different amounts of electricity to each muscle the tension on three string gauges are always the same as each other but one muscle becomes shorter and the other muscle becomes longer.

He finds that he can adjust the ratio of length of the muscle attached to the left wall and the muscle attached to the right wall by adjusting their electrical output.

He finds that when the ratio of length of the left muscle and the right muscle remains constant there is a minimum and maximum amount of force he can create for all three string gauges to identically experience.  He finds that the farther away from a ratio of 1 he is the higher the minimum amount of force and the lower the maximum amount of force he can produce.  And that the closer to a ratio of 1 he is the lower the minimum amount of force and the higher the maximum amount of force he can produce.

Conclusion : 

Adjust experiment three in the following way to make it represent two single joint muscles, two joints and three bones

Replace the string gauge in the middle with a bone and replace the two walls to the left and right with bones.

Replace the length of each muscle with a function of the angle of each joint

Replace the left wall, left muscle and center string gauge with a joint 

Replace the right wall, right muscle and right string gauge with the other joint

In some cases the maximum or minimum angle of one joint may be a function of the position of the other joint because the longer the muscle length relative to resting length the lower the maximum force it can produce and also the shorter the muscle length relative to resting length the lower the maximum force it can produce.

If a muscle can not produce a value between it's maximum and minimum force at a specific muscle length then the muscle can not be at that muscle length

If changing the position of the other joint the muscle does not cross forces the muscle that does not cross the joint to be at a amount of force less than the minimum for a certain muscle length or greater than the maximum for a certain muscle length then that muscle can not exist at that specific muscle length when the joint the muscle does not cross is in that specific position.

This means that the position of a joint a muscle does not cross in some cases can control the range of motion of a joint a muscle does cross and control the minimum and maximum length of a muscle that does not cross the joint.  At least according to this thought experiment

Copyright Carl Janssen 2022 August 8

Additional Reading with useful graphs

http://web.archive.org/web/20220814032815/https://www.houseofhypertrophy.com/the-length-tension-relationship/